Modal identification method for non-proportionally damped structures based on extended sparse component analysis

A modal identification method for non-proportionally damped structures based on extended sparse component analysis, wherein the steps are as follows:

• step 1:
• constructing the analytical signals of the accelerations
• the acceleration of t instant is sampled as x(t)=[x1(t), x2(t), . . . , xm(t)]T, where m is a number of sensors;
• [⋅]T represents the transpose of a matrix;
• the acceleration x(t) is transformed to {circumflex over (x)}(t) by Hilbert transform, which is represented by
• x
• ^
• ⁡
• (
• t
• )
• =
• 1
• π
• ⁢
• ∫
• -
• ∞
• +
• ∞
• ⁢
• x
• ⁡
• (
• τ
• )
• t
• -
• τ
• ⁢
• d
• ⁢
• ⁢
• τ
• ;
• an analytical signal of the acceleration is constructed by {tilde over (x)}(t)=x(t)+j{circumflex over (x)}(t), where j is the imaginary unit;
• step 2:
• obtaining the time-frequency representation of analytical signals
• the analytical signal {tilde over (x)}(t) of the acceleration is transformed into time-frequency domain using STFT;
• a time-frequency domain signal is represented as {tilde over (x)}(t,f)=[{tilde over (x)}1(t,f), {tilde over (x)}2(t,f), . . . , {tilde over (x)}m(t,f)]T and f is frequency index;
• step 3:
• detecting SSPs using average correlation coefficients;
• a range of frequencies regarding a certain time period tk in the time-frequency plane are denoted as an analysis zone F;
• the correlation coefficient between the time-frequency coefficients of two sensor locations a and b is calculated using
• R
• ab
• ⁡
• (
• t
• k
• ,
• F
• )
• =
• r
• ab
• ⁡
• (
• t
• k
• ,
• F
• )
• r
• aa
• ⁡
• (
• t
• k
• ,
• F
• )
• ⁢
• r
• aa
• ⁡
• (
• t
• k
• ,
• F
• )
• ,
• where
• r
• ab
• ⁡
• (
• t
• k
• ,
• F
• )
• =
• ∑
• f
• ∈
• F
• ⁢
• 
• x
• ~
• a
• ⁡
• (
• t
• k
• ,
• f
• )
• ·
• x
• ~
• b
• ⁡
• (
• t
• k
• ,
• f
• )
• 
• ,
• ∀a,b∈{1, . . . , m};
• the correlation coefficients calculated from all sensor locations are averaged to get an average correlation coefficient
• R
• _
• ⁡
• (
• t
• k
• ,
• F
• )
• =
• 1
• M
• ⁢
• ∑
• a
• =
• 1
• m
• ⁢
• ∑
• b
• =
• 1
• ,
• b
• ≠
• a
• m
• ⁢
• R
• ab
• ⁡
• (
• t
• k
• ,
• F
• )
• where M=Cm2;
• Cm2 represents the number of combination forms when picking two different unordered samples from m samples;
• the SSP detection criterion is denoted as R(tk,F)≥1−ε, where ε is a threshold value;
• if the zone (tk,F) satisfies the detection criterion, the points in this zone are all marked as SSPs;
• the detected SSPs are marked as (t,f);
• step 4:
• removing the local outliers of the SSPs through local outlier factor method
• a first sensor location is chosen as the reference;
• the phase differences between each sensor location and the first sensor location are calculated by θ(t,f)=[θ1(t,f), . . . , θm(t,f)], where θk(t,f)=θ{tilde over (x)}1(t,f)−θ{tilde over (x)}k(t,f), ∀k∈{1, . . . , m};
• then the local outlier factors for the phase difference vector at (t,f) are calculated using local outlier factor method;
• the SSPs where the local outlier factors exceed 1 are removed;
• the updated SSPs are denoted as ({circumflex over (t)},{circumflex over (f)});
• step 5:
• clustering the updated SSPs to obtain the amplitudes of the complex-valued mode shapes
• the amplitudes of the time-frequency coefficients at ({circumflex over (t)},{circumflex over (f)}) are grouped by the Hierarchical clustering method;
• the clustering centers are the amplitudes of each complex-valued mode shape which are denoted as |Φ|=[|φ1|, |φ2|, . . . , |φn|];
• Φ is the modal matrix and φi, i=1, . . . , n is the mode shape vector;
• n is the number of modes;
• each group of SSPs after clustering are denoted as ({circumflex over (t)}1,{circumflex over (f)}1), . . . , ({circumflex over (t)}n,{circumflex over (f)}n);
• step 6:
• averaging the phases of each grouped SSPs
• the phase differences for each grouped SSPs are obtained as θ({circumflex over (t)}1,{circumflex over (f)}1), . . . , θ({circumflex over (t)}n,{circumflex over (f)}n);
• the mean values of phase differences for each grouped SSPs are calculated and denoted as θ({circumflex over (t)}1,{circumflex over (f)}1), . . . , θ({circumflex over (t)}n,{circumflex over (f)}n);
• step 7:
• assembling the complex-valued mode shapes using the estimated amplitudes and phases of the mode shapes
• each row of the modal matrix is normalized by dividing the first row to obtain the normalized matrix |Φ|=[|φ1|, |φ2|, . . . , |φn|];
• finally, the complex-valued modal matrix is estimated as Φ=[|φ1|ejθ({circumflex over (t)}1,{circumflex over (f)}1), |φ2|ejθ({circumflex over (t)}2,{circumflex over (f)}2), . . . , |φn|ejθ({circumflex over (t)}n,{circumflex over (f)}n)].